CLASSIFICATION (DISCRIMINATION)

Transcription

1 CLASSIFICATION (DISCRIMINATION) Caren Marzban marzban Generalities This lecture relies on the previous lecture to some etent. So, read that one first. Also, as in the previous lecture, what ou are currentl reading is an epanded version of what is presented at the lecture. Recall, that the problem at hand is to estimate the weights that parameterize the relationship between a set of input variables and a set of output variables (or targets). Actuall, we have to decide on the parameterization, too, since the map also depends on H, the number of hidden nodes. Attributes Features Inputs () Predictand Target (t) Outputs () Map Function Face Name Relation In the previous lecture we considered the regression problem, where the targets are continuous variables. Now, we will see how to handle the case where the targets are discrete, with each value of the target representing some class of objects. For eample, a tornado classification problem could consist of finding the map that relates some set of input variables to, sa, two classes - the class of non-tornadoes, and the class of tornadoes. This wa we can develop a model that can predict whether a set of inputs values corresponds to a tornado or not. To be more precise, the map or the function we are looking for in a classification problem is the (decision) boundar that best demarcates the regions of classes. The figure below shows eamples of a linear boundar (left) and a nonlinear boundar (right) for a noise-free problem.

2 Discriminant Analsis A traditional classification model, called Discriminant Analsis, is an enlightening ancestor of NNs. Let s look at the simple case of one input variable,, and two classes labeled as i =, (e.g., nontornado, and tornado). The model begins b assuming that the likelihood of, when it happens to belong to class=i, is gaussian. In other words, the conditional probabilit of, given class=i, is written as L i () e ( µ i) σ i prob( C = i). where µ i and σ i are the mean and standard deviation of when it belongs to the i th class. Both are estimated from a training set. Note that L + L. For forecasting purposes, this likelihood is the wrong probabilit to look at. We are not interested in the probabilit of finding, sa wind speed = 5 Km/hr, given that there is a tornado on the ground. We want to know the probabilit that there is a tornado on the ground, given that wind speed = 5 Km/hr. This other probabilit is called the posterior probabilit, and the laws of probabilit impl that it is given b p i L i () P i () = p L () + p L (), where p, p are called prior (or climatological) probabilities for the two classes. The are estimated as N /N, and N /N, respectivel, where N i is just the sample size of the i th class, and N = N + N is the total sample size. Note that p + p = and P + P =. Then, when we observe some value of, if P () > P (), then we forecast (or classif) it as a (i.e., a tornado). Actuall, instead of P () > P (), it s more convenient to look at log(p ()/P ()) >, because we can then write the left-hand side as log(p ()/P ()) = D (), where the so-called discriminant function, D (), is given b D () = ( σ ) ( µ σ σ µ ) + ( µ σ σ µ ) + log( σ ) log( p ). σ σ p This equation follows in a straightforward wa b just combining the above equations (homework). Again, an observation from either training set or validation set is forecast as (or assigned to) class (e.g., tornado), if D () >, otherwise it is classified (forecast) as a (e.g., non-tornado). In spite of its simplicit, this is a ver powerful classification model. But as seen from the last equation, the best it can do is to handle quadratic (i.e., ) decision boundaries. It will fail if the boundar underling our data is more nonlinear than that. Of course, the source of this limitation is the starting assumption - normalit. I urge ou to consider using this model, because ou ma be surprised b how well it does. But even if ou never see or hear about this model again, one lesson that ou should not forget is that the distribution of the data is related to the nonlinearit of the boundar. In the above eample, if the distributions are gaussian, then the boundar is quadratic. As a special case, if the distributions are gaussian and equivariant (i.e., σ = σ ), then the boundar is linear. All of this should be reminding ou of the close relationship between gaussian noise and mean square error in linear regression (previous lecture).

3 The following figures show all the above ingredients in both the linear and nonlinear case. Just stud it a bit; I ll go over it during the lecture. It will even be in color then L L 4 L Disc. func. Disc. func. L P P P P NN Discriminant Analsis provides a stepping stone for moving on to classification NNs. First, notice that P () can be re-written in the form (homework) P () =. + e discrim. func. But this is the form of the logistic function discussed previousl. In fact, for the case where the discriminant function is linear, it is eactl the logistic function. And that is nothing but an NN with no hidden nodes. So, ou can see that an NN with hidden nodes - H N in (, ω, H) = g ω i f( ω ij j θ j ) ω, i= is a generalization of Discriminant Analsis. Then, ou should not be surprised to find out that an NN with enough hidden nodes can fit an nonlinear boundar. Additionall, ou should not be surprised to find out that the output of an NN represents posterior probabilit. To strengthen the probabilistic interpretation of NN outputs, in a beautiful paper, Richard & Lippmann (Neural Computation,, 46-8, 99) showed that Neural network classifiers estimate Baesian a- posteriori probabilities. In fact, that s the title of their paper. The eact statement is this: If the activation functions f and g are logistic, then the minimization of cross-entrop - where the target takes values t =,, guarantees j= E = N [t log (, ω) + ( t) log( (, ω))], (, ω) = P () (= prob(c = )). This is a ver important result because it tells ou what ou need to do in order to have an NN that makes probabilistic forecasts. Again, this result should remind ou of the analogous regression result - that the minimization of MSE implies that the NN outputs will be conditional averages: (, ω) =< t >. So, whatever NN simulation ou emplo to perform classification, instruct it to use cross-entrop as the error function, and to use logistic function for all the activation functions. B the wa, for more

4 than two classes, the generalization of the logistic activation function that preserves the probabilistic interpretation of the outputs is called the softma activation function. Method As I said, we still have to worr about all the things we worried about in the regression lecture: Weightdeca, local minima, bootstrap (or something like it), etc. Then, fill in the big table in the previous lecture with different Seed and Bootstrap Trials. The table will allow ou not onl to identif the optimal number of hidden nodes, but also to place error bounds on the performance of the NN. One difference with the regression NN is that the choice of the error function is a little less ad hoc, in the sense that onl cross-entrop (with logistic activation functions) will ield probabilistic outputs. Another difference is that ou ma have to worr about the representation of the different classes in the training set. For eample, if one of the classes is drasticall under-represented in the training set (compared with the other classes), then the NN ma just treat the cases in that class as noise. It is this concern that often prompts some people to balance the classes in the training set. This is actuall not recommended, because the outputs will no longer represent posterior probabilities. For the outputs to have that probabilistic interpretation, it is crucial for the different classes in the training set to be represented eactl proportional to their respective prior (or climatological) probabilities. So, it s better not t dabble with the class sizes. Having said that, there will be times that the above concern will in fact be realized. In that case, it is possible to practice that balancing act, if ou assure that the outputs are scaled in such a wa to resurrect their probabilistic interpretation. In particular, ou would have to multipl each output node b the prior probabilit of the class designated b that output node, and then renormalize to assure that all the probabilities add up to. One final difference is in assessing the performance of a classification NN, especiall if the outputs are arranged to be posterior probabilities. Because then ou are dealing with probabilistic forecasts, and if ou are a meteorologist ou probabl know that the verification of probabilistic forecasts is a topic that requires a whole book. I ll discuss some of the important concepts below, and in m net lecture. But at least check out the following two classic papers b Murph, A. H., and R. L. Winkler: Int. J. Forecasting, 7, (99), and Mon. Wea. Rev., 5, -8 (987). Wilks book is good to have as well (Wilks, D. S., 995: Statistical Methods in the Atmospheric Sciences. Academic Press, NY.) 4

5 Overfitting In practice, we still have to worr about all the nuisances we talked about for the regression NN. Overfitting (or the optimal number of hidden nodes) is the first concern. The following figures illustrate overfitting at work. I have concocted a nois version of the classification eample I showed in the beginning of the lecture. The boundar is the same inverted Meican hat as before (labeled with light circles), but it s nois in the sense that a few data points are on the wrong side of the boundar. Instead of and, I ve used filled and unfilled triangles. So, some filled triangles are above the boundar and some unfilled triangles are below the boundar. I ve trained a series of NNs with different number of hidden nodes: H =, 4, 6, 8,, and 5. The corresponding decision boundar is shown with the dark line. You can see that the H = NN underfits the boundar. H = 4 does better, but H = 6 is alread overfitting in the sense that it thinks the single filled triangle at the top of the upper region is a real data point, rather than noise, which it clearl is. Larger NNs, with H = 8,, and 5, just continue to overfit even more. So, H = 4 is optimal. H = H = H = 6 H = H = H =

6 Practical Application - Hail Size 5 Frequenc Hail Size (mm) Treating hail size as a continuous quantit, we developed an NN that predicts hail size. Given the peaks in the distribution of hail size (above), it is natural to develop an NN that predicts the size class to which a set of input values belong. In fact, it would be useful to find the probabilit of getting hail size of a certain class? The reason for the three peaks in the distribution is mostl human related. It turns out there are two tpes of meteorologists - those who report hail size in millimeters (or hundredth of an inch), and those who classif hail as being coin size, golf ball size, or baseball size. The preprocessing for a classification NN is mostl the same as for a regression NN. Ecluding collinear inputs, removing outliers, transforming to z-scores, etc. One difference is that consideration has to be given to the class. For instance, instead of eamining the distribution of input variables (like we did in the regression NN), we must look at the class-conditional distribution of the input variables. That means that ou must plot the distribution/histogram of the input variables separatel for each class. The figure below shows this for one of the input variables, for the classes of hail size. As epected, the distributions are bell-shaped, and shift to the right as the size of the corresponding class increases. 5 Frequenc Input Variable 6

7 Similarl, in identifing collinear inputs, instead of calculating the r of two inputs, ou should look at the r of two inputs for each class separatel. It s possible that two inputs are highl collinear when the belong to one class, but not at all collinear in another class. Clearl, ou would not want to eclude a member of such a pair, at least not based on their collinearit. Going through the previousl outlined method, we find the following optimal values: The number of inputs nodes = 9, and the number of hidden nodes = 4. The number of output nodes is a bit trick; it depends on how we choose to encode the class information in the target nodes. For instance, it s possible to have one output node, with values,.5, and representing the classes. And if the three classes have an order associated with them, then that s probabl fine. A better wa - and one that preserves the probabilistic interpretation of the outputs - is to use what s called -of-c encoding. In that scheme, there is one output node for each class, and the larger one designates the class. During training, for class cases, the three output nodes are assigned the values (,,) as their target values; class cases are assigned (,,) as their corresponding output nodes, etc. Performance This section is a little long, so make ourself comfortable. And, no, it could not have been placed as an appendi because its content is crucial for properl assessing the performance of an NN. Performance measures come in a variet of tpes, depending on the tpe of forecast (and observation). For eample, if the forecasts are categorical, then one ma have a scalar measure of performance, or a multidimensional measure (like a diagram). Similarl, probabilistic forecasts can be assessed in terms of scalars and/or diagrams. The following table lists an eample of each tpe. Scalar Multi-dimensional Categorical Equitable Threat Score ROC Diagram Probabilistic Ranked Probabilit Score Attributes Diagram ROC stands for Receiver s Operating Characteristic, and will be defined below. Needless to sa, multidimensional measures carr more information than scalar ones; similarl, probabilistic ones contain more information than categorical ones. Then again, the all have their place in the world. At one etreme we have categorical forecasts and observations, like hail classes being classified (forecast) into classes. The starting point for assessing the performance for such a classifier is the contingenc table: C-table = n n n n n n n n n where n is the number of class cases (incorrectl) classified as belonging to class. Etc. In m (nonuniversal) convention, the rows correspond to observations, and the columns represent forecasts. There is an infinit of was to construct scalar measures from the contingenc table, but almost no multidimensional measures. B contrast there are numerous multidimensional measures that can be constructed from a contingenc table. For that reason, the problem of classifing three classes - labeled,, and - is broken down to three -class problems: classification of class or otherwise, class or otherwise, and class or otherwise. Although this treatment of a -class problem can lead to some loss of information, the loss is partiall compensated b the availabilit of multidimensional measures. Then, performance can be represented with three contingenc tables: ( ) n n C-table =, n n 7,

8 A couple of common measures derived from this table are are the Probabilit of Detection and False Alarm Rate (not Ratio - that s something else). If the class we are tring to detect is the one labeled, then POD = n n + n, FAR = n n + n. Note that the total number of class cases is given b N = n + n, and that of class cases is N = n + n. The total sample size is N = N + N. Even though, our NN produces probabilistic forecasts, it is possible to categorize the forecasts b thresholding them; ou simpl pick a number between and (because probabilit lives in that range) and then sa that an forecast less than that number belongs to class, and an forecast larger than the threshold belongs to class. In this wa, one can reduce a continuous quantit like probabilit into a categorical one. Then, the question is which threshold to pick. The ROC diagram handles that problem. In fact, an ROC diagram is a parametric plot of POD versus FAR, as the threshold is varied from to. So, ou pick a threshold, but ou don t have to stick to it; ou var it from to in some increments, calculating POD and FAR at each increment, and then plotting them on an - plot. It is eas to show that a classifier with no abilit to discriminate between two classes ields a diagonal line of slope one and intercept zero; otherwise it bows above the diagonal line. (The area under the curve is often taken as a scalar measure of the classifier s performance, and so, a perfect classifier would have an area of under its ROC curve, while a random classifier would have an area of.5. In addition to its multidimensionalit, another virtue of a ROC diagram is its abilit to epress performance without a specific reference to a unique threshold. A specific choice of the threshold calls for knowledge of the costs of misclassification which are user dependent. In this wa, a ROC diagram offers a user-independent assessment of performance. For the Hail problem, the ROC plots are shown below. The left diagram is for the training set and the right one is for the validation set. (The error bars are the standard error from the bootstrap trials.) It can be seen that performance of class and class forecasts is quite high (given b the amount of bowing awa from the diagonal). However, class forecasts are not doing too well, since according to the validation ROC plot, the are almost as good as random (evidenced b the overlap of the error bars and the diagonal line). So, the NN does good in predicting the largest and the smallest of the hail classes (in that order), but not too well with the midsize class. If ou think about it, it makes sense that the midsize class would be the hardest one to discriminate from the others..8.8 Probabilit of Detection.6.4 Probabilit of Detection False Alarm Rate False Alarm Rate Of course, it is possible to assess the qualit of the forecasts without categorizing the forecasts. Murph and Winkler (referenced, above) have shown us how to do that. The show that the various aspects of the qualit of probabilistic forecasts can be captured b three diagrams - reliabilit (or calibration), refinement, and discrimination diagrams. (A slightl improved version of the reliabilit diagram is an 8

9 attributes diagram; it shows whether or not a given forecast actuall contributes to the skill of the classifier). These are all computed from the joint probabilit of forecasts and observations, but I ll give a simpler description here. I ll return to the joint distribution in the net lecture. A reliabilit diagram displas the etent to which the frequenc of an event at a given forecast probabilit matches the actual forecast probabilit. To construct it, take all the cases that have a forecast of %. Mark this number on the -ais. What fraction of the whole data is the number of cases in this set? Mark that answer on the -ais. We now have one point on the - plane, which is hopefull somewhere near the diagonal, because a % probabilit is supposed to mean that % of the time we get that outcome. Then repeat this for forecasts that are %, %,..., 9%. All of the resulting points should lie somewhere near the diagonal; the closer to the diagonal, the more reliable the forecasts. A refinement diagram displas the sharpness of the forecasts, i.e., the etent to which ver high or ver low probabilities are issued. To construct it, just plot the histogram of all the forecasts. There are man was in which a classifier can be poor in terms of refinement. For eample, a classifier that produces onl two forecasts, sa % and 9%, has poor refinement. A good classifier would produce a range of all forecasts (between and %). I put reliabilit and refinement plots on the same diagram (below). Again, the error bars are the standard error from the bootstrap trials. The diagonal line corresponds to perfect reliabilit. We can see that within the error bars the NN produces forecasts that are perfectl reliable for all three classes. The curve (blue) not along the diagonal is the refinement plot. The class forecasts have a ver reasonable refinement plot in that the are relativel flat in the full range of to %. The refinement plot for class forecasts suggests that the never eceed 6%; below 6% the are reasonabl refined. Class forecasts are reasonabl refinement as well; the peak at % reflects the fact that class cases are rare. The shaded regions are what make these attributes diagrams. See Wilks book (referenced above) for details..8.8 Fraction Observed.6.4 Fraction Observed Class Forecast Probabilit Class Forecast Probabilit.8 Fraction Observed Class Forecast Probabilit 9

10 A discrimination plot ehibits the etent to which the forecasts discriminate between the classes. To construct it, plot the class-conditional distributions (or histograms) of the forecasts. The top figure (below) is almost a tet-book case of what a good classifier s discrimination plot should look like. When class cases are presented to the NN, it produces class forecasts that peak at higher probabilities. At the same time, if class cases are presented to the NN, the class forecasts are mostl in the lower probabilities. Class cases occup the middle ground. Skipping to the bottom diagram, the behavior is almost the complement of that seen in the top diagram. Class forecasts are more frequent when class cases are shown, etc. Finall, in the middle diagram, it can be seen that classes and are reasonabl discriminated b class forecasts. Same with class and. But classes and are not at all discriminated b class forecasts. This, again, points to the problem the NN is having in classifing class cases Frequenc Class Forecast Probabilit Frequenc Class Forecast Probabilit.8.6 Frequenc Class Forecast Probabilit Conclusion We are done. M goal was to teach ou about regression and classification as done with NNs. Although I have covered onl the tip of the iceberg, it should give ou enough to pla with and to get our hands dirt. In the process, we developed a number of NNs for estimating hail size from some measurable attributes. We even ended-up using the same data set for developing both a regression and a classification NN for that task. Together the provide an output like this: Forecast hail size = 4mm, Prob. of coinsize hail = 8%, Prob. of golf ball-size hail = 5%, Prob. of baseball-size hail = 5%. M other goal was to address the issue of performance (or verification). In m eperience, performance does not get nearl the attention it deserves. After all, what s the point of developing a high-powered NN if we don t have a proper wa of assessing if it is indeed high-powered? As if the performance assessment of categorical forecasts is not comple enough, probabilistic forecasts make the business even more comple. The source of the compleit is reall the multidimensionalit of performance itself. But don t loose hope; the situation is relativel under control, and a little reading in the literature should confirm that. Just remember that because of the multidimensionalit of performance, it s better not to quantif performance too much, because quantification usuall requires distillation, which in turn implies loss of information.